integral of cos^6(θ)sin^2(θ)

解

\int cos^{6} (θ) sin^{2} (θ) d θ

\frac{5}{192} cos^{3} (θ) sin (θ) + \frac{1}{6} cos^{5} (θ) sin (θ) + \frac{5}{32} (2 θ + sin (2 θ)) - \frac{1}{8} cos^{7} (θ) sin (θ) - \frac{7}{48} cos^{5} (θ) sin (θ) - \frac{35}{256} (2 θ + sin (2 θ)) + C

解答ステップ

\int cos^{6} (θ) sin^{2} (θ) d θ

三角関数の公式を使用して書き換える

= \int cos^{6} (θ) (1 - cos^{2} (θ)) d θ

拡張

cos^{6} (θ) (1 - cos^{2} (θ)) : cos^{6} (θ) - cos^{8} (θ)

= \int cos^{6} (θ) - cos^{8} (θ) d θ

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int cos^{6} (θ) d θ - \int cos^{8} (θ) d θ

\int cos^{6} (θ) d θ = \frac{sin ( θ ) cos ^{5} ( θ )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (θ) sin (θ) + \frac{3}{8} (θ + \frac{1}{2} sin (2 θ)))

\int cos^{8} (θ) d θ = \frac{sin ( θ ) cos ^{7} ( θ )}{8} + \frac{7}{8} (\frac{sin ( θ ) cos ^{5} ( θ )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (θ) sin (θ) + \frac{3}{8} (θ + \frac{1}{2} sin (2 θ))))

= \frac{sin ( θ ) cos ^{5} ( θ )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (θ) sin (θ) + \frac{3}{8} (θ + \frac{1}{2} sin (2 θ))) - (\frac{sin ( θ ) cos ^{7} ( θ )}{8} + \frac{7}{8} (\frac{sin ( θ ) cos ^{5} ( θ )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (θ) sin (θ) + \frac{3}{8} (θ + \frac{1}{2} sin (2 θ)))))

簡素化

\frac{sin ( θ ) cos ^{5} ( θ )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (θ) sin (θ) + \frac{3}{8} (θ + \frac{1}{2} sin (2 θ))) - (\frac{sin ( θ ) cos ^{7} ( θ )}{8} + \frac{7}{8} (\frac{sin ( θ ) cos ^{5} ( θ )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (θ) sin (θ) + \frac{3}{8} (θ + \frac{1}{2} sin (2 θ))))) : \frac{5}{192} cos^{3} (θ) sin (θ) + \frac{1}{6} cos^{5} (θ) sin (θ) + \frac{5}{32} (2 θ + sin (2 θ)) - \frac{1}{8} cos^{7} (θ) sin (θ) - \frac{7}{48} cos^{5} (θ) sin (θ) - \frac{35}{256} (2 θ + sin (2 θ))

= \frac{5}{192} cos^{3} (θ) sin (θ) + \frac{1}{6} cos^{5} (θ) sin (θ) + \frac{5}{32} (2 θ + sin (2 θ)) - \frac{1}{8} cos^{7} (θ) sin (θ) - \frac{7}{48} cos^{5} (θ) sin (θ) - \frac{35}{256} (2 θ + sin (2 θ))

定数を解答に追加する

= \frac{5}{192} cos^{3} (θ) sin (θ) + \frac{1}{6} cos^{5} (θ) sin (θ) + \frac{5}{32} (2 θ + sin (2 θ)) - \frac{1}{8} cos^{7} (θ) sin (θ) - \frac{7}{48} cos^{5} (θ) sin (θ) - \frac{35}{256} (2 θ + sin (2 θ)) + C